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Questions tagged [finite-element]

A means of solving ordinary and partial differential equations. The domain of the problem is broken up into elements, and the solution in each element is expanded in a basis of functions. The Finite Element Method lends itself well to adaptive refinement, irregular geometry, and good error estimates.

A means of solving ordinary and partial differential equations through discretization of the defining differential equations into elements. Individual element mass and stiffness matrices are assembled and from them global stiffness and mass matrices are assembled and used to solve the coupled system. The Finite Element Method lends itself well to adaptive refinement, irregular geometry, and good error estimates.

The methods discretize a complex physical system as a set of many elements having a very simple geometrical shape (e.g. a triangle-shaped plate), for which it it is simple to solve the differential equation. A global mass and stiffness matrix are then formed from the element matrices and used to solve the system

See Wikipedia page on FEMs.

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Basic explanation of shape function

I just started studying FEM in a more structured basis compared to what I used to do during my undergraduate courses. I am doing this because, despite the fact that I can use the "FEM" in commercial (and other non-commercial) software, I would like…
Alfonso Santiago
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What is the purpose of the test function in Finite Element Analysis?

In the wave equation: $$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$ Why do we first multiply by a test function $v(x,t)$ before integrating?
Andy
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Impose the compatibility conditions for mixed finite elements method in Stokes equation

$\newcommand{\v}[1]{\boldsymbol{#1}}$ Suppose we have following Stokes flow model equation: $$ \tag{1} \left\{ \begin{aligned} -\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f} \\ \mathrm{div} \v{u} &= 0 \end{aligned} \right.$$ where the viscosity…
Shuhao Cao
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Mathematically, why does mass matrix / load vector lumping work?

I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented a code where the load vector is assembled in a lumped fashion rather than an FEM-consistent fashion. But I've never looked…
Paul
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FEM: singularity of the stiffness matrix

I'm solving the differential equation $$ \left( \sigma^{2}(x) u ''(x) \right)'' = f(x), \;\;\; 0 \leqslant x \leqslant 1 $$ with initial conditions $u(0) = u(1) = 0$, $u''(0) = u''(1) = 0$. Here $\sigma(x) \geqslant \sigma_{0} > 0$ is…
Appliqué
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Who uses finite elements with higher continuity?

Lagrange elements of any polynomial describe piecewise continuous functions. Typically, those functions are differentiable. Mixed finite element methods use vector fields of even less continuity, such as normal continuity. With some great…
shuhalo
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Standard format for finite element meshes

Does there exist a standard format for finite element meshes which is widely used in the industry? Thanks!
Benjamin
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On the completeness of the Periodic Table of Finite Elements

In a recent SIAM News article, there is a long article describing a systematic organization of the finite elements, aptly dubbed the Periodic Table of Finite Elements. Its really quite fascinating to see how classification can be accomplished via…
Paul
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A simple question about 1D finite element derivatives

For 1D derivative we have \begin{equation} F(x) = \frac{\partial f(x)}{\partial x} \end{equation} \begin{equation} f(x)=\sum_{i}f_ie_i(x) \end{equation} \begin{equation} F(x)=\sum_{i}F_ie_i(x) \end{equation} where…
yangjinhui
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Reference implementation of Nédélec-Elements

Does anybody know of an implementation of Nédélec elements that does not come along with a huge bulk of additional software? Is there a small library written in a language like Python, Matlab, or Octave? The only implementations I know of are part…
shuhalo
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Appropriate space for weak solutions to an elliptical pde with mixed inhomogeneous boundary conditions

I'm working with the following mixed inhomogeneous boundary value problem: $\nabla(\kappa\nabla u)=f$ in $\Omega$ with $\partial\Omega = \Omega_1 \bigcup\Omega_2$ such that $u=g$ on $\partial\Omega_1$ $\kappa\nabla u\cdot n = h$ on $\Omega_2$ …
Paul
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implementing higher order derivatives for finite element

I am implementing higher order derivatives for FEM. Example, to solve a Poisson problem, biharmonic or triharmonic PDE one needs first, second or third order derivatives respectively. As usually done in FEM, there is a mapping from a reference…
uli.xu
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Reconstructing a continuous function from finite element method. Is there a faster algorithm for doing so?

Lets say I've decomposed a continuous function $y(x)$ over some domain $L_x$ using known finite element method with local basis $Q_i(x)$. Suppose $L$ is divided into $M$ "elements". If I want to know the function $y(x)$ at a point $x=p$ (where $p$…
user2350366
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Forced viscous damping in elastodynamics

I have an 2D elastodynamics problem, that is a problem which is driven by the Cauchy equation: $$\rho\ddot u-\mathrm{div}\sigma=\rho f$$ where $u$ is the displacement, $\sigma$ the Cauchy stress tensor, $f$ the density of volumetric forces, and…
Benjamin
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What exactly causes mesh locking in thin plate bending problems?

In thin plate bending problems, it would be very nice if we could model a thin rectangular plate with a mesh of many elements in the transverse directions, but only a single element in the longitudinal direction. However, when try to model a plate…
Paul
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